G01DHF : NAG Library, Mark 24

3 Description

G01DHF computes one of the following scores for a sample of observations,

x_{1}, x_{2}, \dots, x_{n}

Rank Scores
The ranks are assigned to the data in ascending order, that is the $i$ th observation has score $s_{i} = k$ if it is the $k$ th smallest observation in the sample.
Normal Scores
The Normal scores are the expected values of the Normal order statistics from a sample of size $n$ . If $x_{i}$ is the $k$ th smallest observation in the sample, then the score for that observation, $s_{i}$ , is $E (Z_{k})$ where $Z_{k}$ is the $k$ th order statistic in a sample of size $n$ from a standard Normal distribution and $E$ is the expectation operator.
Blom, Tukey and van der Waerden Scores
These scores are approximations to the Normal scores. The scores are obtained by evaluating the inverse cumulative Normal distribution function, $Φ^{- 1} (\cdot)$ , at the values of the ranks scaled into the interval $(0, 1)$ using different scaling transformations.

The Blom scores use the scaling transformation $\frac{r_{i} - \frac{3}{8}}{n + \frac{1}{4}}$ for the rank $r_{i}$ , for $i = 1, 2, \dots, n$ . Thus the Blom score corresponding to the observation $x_{i}$ is
$s_{i} = Φ^{- 1} (\frac{r_{i} - \frac{3}{8}}{n + \frac{1}{4}}) .$

The Tukey scores use the scaling transformation $\frac{r_{i} - \frac{1}{3}}{n + \frac{1}{3}}$ ; the Tukey score corresponding to the observation $x_{i}$ is
$s_{i} = Φ^{- 1} (\frac{r_{i} - \frac{1}{3}}{n + \frac{1}{3}}) .$

The van der Waerden scores use the scaling transformation $\frac{r_{i}}{n + 1}$ ; the van der Waerden score corresponding to the observation $x_{i}$ is
$s_{i} = Φ^{- 1} (\frac{r_{i}}{n + 1}) .$

The van der Waerden scores may be used to carry out the van der Waerden test for testing for differences between several population distributions, see Conover (1980).
Savage Scores
The Savage scores are the expected values of the exponential order statistics from a sample of size $n$ . They may be used in a test discussed by Savage (1956) and Lehmann (1975). If $x_{i}$ is the $k$ th smallest observation in the sample, then the score for that observation is
$s_{i} = E (Y_{k}) = \frac{1}{n} + \frac{1}{n - 1} + \dots + \frac{1}{n - k + 1},$
where $Y_{k}$ is the $k$ th order statistic in a sample of size $n$ from a standard exponential distribution and $E$ is the expectation operator.

Ties may be handled in one of five ways. Let

x_{t (i)}

, for

i = 1, 2, \dots, m

, denote

m

tied observations, that is

x_{t (1)} = x_{t (2)} = \dots = x_{t (m)}

with

t (1) < t (2) < \dots < t (m)

. If the rank of

x_{t (1)}

k

, then if ties are ignored the rank of

x_{t (j)}

will be

k + j - 1

. Let the scores ignoring ties be

s_{t (1)}^{*}, s_{t (2)}^{*}, \dots, s_{t (m)}^{*}

. Then the scores,

s_{t (i)}

, for

i = 1, 2, \dots, m

, may be calculated as follows:

– if averages are used, then $s_{t (i)} = \sum_{j = 1}^{m} s_{t (j)}^{*} / m$ ;
– if the lowest score is used, then $s_{t (i)} = s_{t (1)}^{*}$ ;
– if the highest score is used, then $s_{t (i)} = s_{t (m)}^{*}$ ;
– if ties are to be broken randomly, then $s_{t (i)} = s_{t (I)}^{*}$ where $I \in \{random permutation of 1, 2, \dots, m\}$ ;
– if ties are to be ignored, then $s_{t (i)} = s_{t (i)}^{*}$ .

4 References

Blom G (1958) Statistical Estimates and Transformed Beta-variables Wiley

Conover W J (1980) Practical Nonparametric Statistics Wiley

Lehmann E L (1975) Nonparametrics: Statistical Methods Based on Ranks Holden–Day

Savage I R (1956) Contributions to the theory of rank order statistics – the two-sample case Ann. Math. Statist. 27 590–615

Tukey J W (1962) The future of data analysis Ann. Math. Statist. 33 1–67

5 Parameters

1: SCORES – CHARACTER(1)Input

On entry: indicates which of the following scores are required.

$SCORES ='R'$: The ranks.
$SCORES ='N'$: The Normal scores, that is the expected value of the Normal order statistics.
$SCORES ='B'$: The Blom version of the Normal scores.
$SCORES ='T'$: The Tukey version of the Normal scores.
$SCORES ='V'$: The van der Waerden version of the Normal scores.
$SCORES ='S'$: The Savage scores, that is the expected value of the exponential order statistics.

Constraint:

SCORES ='R'

'N'

'B'

'T'

'V'

'S'

2: TIES – CHARACTER(1)Input

On entry: indicates which of the following methods is to be used to assign scores to tied observations.

$TIES ='A'$: The average of the scores for tied observations is used.
$TIES ='L'$: The lowest score in the group of ties is used.
$TIES ='H'$: The highest score in the group of ties is used.
$TIES ='N'$: The nonrepeatable random number generator is used to randomly untie any group of tied observations.
$TIES ='R'$: The repeatable random number generator is used to randomly untie any group of tied observations.
$TIES ='I'$: Any ties are ignored, that is the scores are assigned to tied observations in the order that they appear in the data.

Constraint:

TIES ='A'

'L'

'H'

'N'

'R'

'I'

3: N – INTEGERInput

On entry:

n

, the number of observations.

Constraint:

N \geq 1

4: X(N) – REAL (KIND=nag_wp) arrayInput

On entry: the sample of observations,

x_{i}

, for

i = 1, 2, \dots, n

5: R(N) – REAL (KIND=nag_wp) arrayOutput

On exit: contains the scores,

s_{i}

, for

i = 1, 2, \dots, n

, as specified by SCORES.

6: IWRK(N) – INTEGER arrayWorkspace

7: IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ or ​ 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is

0

. When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

7 Accuracy

For

SCORES ='R'

, the results should be accurate to machine precision.

For

SCORES ='S'

, the results should be accurate to a small multiple of machine precision.

For

SCORES ='N'

, the results should have a relative accuracy of at least

\max (100 \times ε, 10^{- 8})

where

ε

is the machine precision.

For

SCORES ='B'

'T'

'V'

, the results should have a relative accuracy of at least

\max (10 \times ε, 10^{- 12})

On entry,	$SCORES \neq'R'$ , $'N'$ , $'B'$ , $'T'$ , $'V'$ or $'S'$ ,
or	$TIES \neq'A'$ , $'L'$ , $'H'$ , $'N'$ , $'R'$ or $'I'$ ,
or	$N < 1$ .

NAG Library Routine Document

G01DHF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine DocumentG01DHF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

G01DHF